Lemma 114.25.7. Let $R$ be a local ring.

1. If $(M, N, \varphi , \psi )$ is a $2$-periodic complex such that $M$, $N$ have finite length. Then $e_ R(M, N, \varphi , \psi ) = \text{length}_ R(M) - \text{length}_ R(N)$.

2. If $(M, \varphi , \psi )$ is a $(2, 1)$-periodic complex such that $M$ has finite length. Then $e_ R(M, \varphi , \psi ) = 0$.

3. Suppose that we have a short exact sequence of $2$-periodic complexes

$0 \to (M_1, N_1, \varphi _1, \psi _1) \to (M_2, N_2, \varphi _2, \psi _2) \to (M_3, N_3, \varphi _3, \psi _3) \to 0$

If two out of three have cohomology modules of finite length so does the third and we have

$e_ R(M_2, N_2, \varphi _2, \psi _2) = e_ R(M_1, N_1, \varphi _1, \psi _1) + e_ R(M_3, N_3, \varphi _3, \psi _3).$

Proof. This follows from Chow Homology, Lemmas 42.2.3 and 42.2.4. $\square$

Comment #2608 by Ko Aoki on

Typo in the statement of (3): "$(2,1)$-periodic complexes" should be replaced by "$2$-periodic complexes."

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).